electrophoresis of a thin charged disk - princeton universitystonelab/publications... ·...

ARTICLES

Electrophoresis of a thin charged disk J. D. Sherwooda) Schlumberger Cambridge Research, High Cross, Madingley Road, Cambridge CB3 OEL, United Kingdom

H. A. Stone Division of Applied Sciences, Harvard University Cambridge, Massachusetts 02138

(Received 31 August 1994; accepted 14 December 1994)

The electrophoretic velocity of a charged disk of zero thickness is computed in the limit of small surface potentials, but with arbitrary double layer thickness. The disk represents an idealized clay particle, and has uniform surface charge over its flat surface, together with a uniform line charge around its edge. The contributions of these two charges to the electrophoretic velocity are considered separately. Asymptotic results are obtained for thin and thick double layers, and intermediate results are obtained by numerical integration. The singularities in both the electrical and hydrodynamic fields at the edge of the particle enhance the importance of the edge charge when the double layer thickness K-I is small compared to the disk radius a. 0 1995 American Institute of Physics.

I. INTRODUCTION

We consider the electrophoresis of a charged circular disk of zero thickness. The charge density (T on the surface of the particle is assumed to be u$form, as is the line charge density q around the circular edge of the disk. The disk represents an idealized plate-like clay particle (e.g., montmorillonite), with a negative surface charge (+ caused by substitu- tions in the crystal lattice of the clay, and an edge charge q which depends upon the pH within the surrounding fluid.

Clays are widely studied because of their importance in agriculture and civil engineering, and because of their use as vicosifiers in industrial products ranging from pharmaceuti- cals to drilling fluids. The state of flocculation (and hence the rheology) of an aqueous suspension of clay particles depends upon the charges on the clay, and it is therefore important to be able to interpret measurements of the electrophoretic velocity of such particles. These-velocities can increase*8 by as much as 50% as the pH increases to values above 10. We shall assume that the charge density cr of the crystal lattice is uniform over the surface of the particle. However, some studies suggest that the electrophoretic velocity is independent of electrolyte concentration, implying that a constant surface potential may be a more appropriate boundary condition. We return to this point in Sec. V.

MontmorilIonite particles have a thickness typically 1 nm and lateral dimensions typically up to 1 pm. If suspended in an aqueous electrolyte, the charged particle will be surrounded by a cloud of counterions. The thickness of the cloud is o(K-l), where the Debye length K-~-IO nm in a l-l electrolyte of concentration 10V3 mol/l. Thus the Debye length is usually small compared to the lateral dimension of the particle, but is not necessarily small compared to the

%Zorresponding author: Telephone: 44 lk3 325363, fax: 44 1223 467004; te-mail: [email protected]

particle thickness. It is, therefore, not evident that Smolu- chowski’s result for the electrophoretic velocity of a particle with a thin double layer is valid. Here we shall study the infinitely thin disk for arbitrary values of UK, where a is the disk radius, and investigate both surface and edge charge distributions.

Previous studies of clay-like particles have been based upon analyses of thin oblate spheroids. Yoon and Kim3 studied spheroids with a uniform surface potential @a (rather than a uniform surface charge density o), with arbitrary double layer thickness K-~. Fair and Anderson4 specified the surface potential +,, as a function of position over the surface of a spheroid, but assumed that the double layer was everywhere thin compared to the particle dimension. Here we shall tackle the disk geometry directly by means of Hankel transforms. We consider a uniform surface charge density CT and a uniform edge charge q, with no restriction on the Debye length K-l.

We assume that the charge cloud can be described by means of the linearized Poisson-Boltzmann equation, and we shall neglect deformation of the charge cloud caused ei- ther by the applied electric field Em or by motion of the fluid. The Reynolds number for flow around such micron-size particles is sufficiently small that inertial effects may ‘be neglected. Our analysis is, therefore, similar to that of Henry, who studied electrophoresis of a uniformly charged sphere. These simplifying assumptions enable us to compute the electrophoretic velocity of a particle by means of the reciprocal theorem,6’7 without the need to compute the details of the electrically driven fluid velocity around the particle. Non- spherical particles become tractable, and this approach has been used to study both rods6 and spheroids3”

The boundary value problem is linear, and we may consider separately the case of an electric field Em normal to the surface of the particle, which causes the particle to translate broadside on, and an electric field parallel to the plane of the

Phys. Fluids 7 (4), April 1995 1070-6631/95/7(4)/697/9/$6.00 6 1995 American Institute of Physics 697

Downloaded 20 Jun 2003 to 140.247.59.174. Redistribution subject to AIP license or copyright, see http://ojps.aip.org/phf/phfcr.jsp

FIG. 1. Disks with uniformly distributed surface and edge charges. AppIication of an electric field parallel or perpendicular to the disk face produces, respectively, edgewise or broadside translation.

particle, which leads to edgewise motion. The total charge on a particle of radius a is &=27i-(a2u+aq). If the charge cloud is very large compared to the size of the disk (i.e., UK~), the cloud is sufficiently diffuse that it may be ig- nored. The electrical force QE” acting on the particle is balanced by the Stokes drag, which for a particle translating in a Newtonian fluid of viscosity ,U is 16pa U&,ad for broadside motion with velocity lJbtoad, and (3 2/3),ua Uedge for edgewise motion with velocity Uedge .

When the particle is surrounded by a charge cloud of counterions, the applied electric field will act on the ions, thereby creating a body force which causes motion of the fluid. This fluid ‘motion will act on the particle, and will reduce the electrophoretic velocity. The basic governing equations for electrokinetic phenomena are available in standard references (e.g., Savilles), and in the following sections we summarize these analyses. Our work is based on (previ- ously published) solutions of the Laplace equation, the linearized Poisson-Boltzmann equation and the Stokes equations, obtained by means of Hankel transforms and summarized in Sec. II, and we deal directly with the disk geometry with its sharp edges rather than studying the limiting case of an analysis for (smooth) oblate spheroids. Ana- lytical estimates for the translational speed for edge and surface charge distributions are determined in ‘Sec. III, for the thin (a~%l) and thick (a~4) double layer limits. Numeri- cal results for 10-1<a~C102 are given in’Sec. IV and are compared with the asymptotic analyses. A discussion of the applicability of this work to real clays is provided in Sec. V

II. ANALYSIS

A. Equilibrium charge cloud

We suppose that the disk, of radius a, is suspended in a Newtonian electrolyte of viscosity b containing m species of ions, with valence zi (i= 1,. . .,m) and number density pli(r), where r denotes position (see Fig. 1). Far away from the surface of the charged disk, the ionic number densities attain bulk values ni. In the absence of any applied field or fluid motion, the ionic densities satisfy the Boltzmann distribution

nb=n~ exp( -ez&alkT), (1) where e is the electronic charge, k is Boltzmamr’s constant, and T is the temperature. The electric potential +,, satisfies the Poisson equation

698 Phys. Fluids, Vol. 7, No. 4, April 1995

v2coE-g !z+, i=l

(2)

where E is the permittivity of the electrolyte. We shall assume that the potentials are small (i.e., e&,/kT<l) so that we may linearize the Poisson-Boltzmann equation obtained by combining (1) and (2). This linearization leads to

v2h=K2’$o, (3)

where the Debye length K-~, which characterizes the thickness of the charge cloud, is defined by

m e’z?n’ K2=C 1.

i=l ekT (4)

The local ion charge density p- Xr=“,,ez$ becomes, in the limit e &,lkTel,

po(r)=-c2f$0(r). (5)

For the case of a disk with uniform surface charge density a; the potential h in the upper half-plane z&O satisfies the boundary conditions

840 Edz=-a’

z=o+, Osr<a, (64

GO E x=0, z=O, r>a, (6b)

40-4 Irl-+~. (64 The solution to the linear Poisson-Boltzmann equation (3) satisfying the boundary conditions (6a)-(6c), may be obtained by means of Hankel transforms:

q50( r,z) = 3 1: J~$~:,::i~ exp[ - ( K2 + t2) 1’2Z]dt

(7)

where the Ji are Bessel functions and Qsti is the total surface charge on the disk; (7) defines the function c$~. If a~sl,’ and z+a, the potential will be that above an un- bounded, charged plane

+o(r,z) = CT exp( - KZ)/EK (8)

in the region a - r9K-1 away from the edge of the disk.

J. D. Sherwood and I+ A. Stone


For the case of a uniform edge (or ring) charge with line density 4, the boundary conditions (6a)-(6b) must be re- placed by

a% -=-;*6(‘-a) f dz z=o ,

and this leads to the potential

= 2 &(r,z). 00)

Note that the potential will become large close to a line charge, with a logarithmic singularity. In practice, if the-line charge density. q is too high, counterion condensation can occur.lo*ll The limiting line charge density q=4rrekTle, corresponds to a line of isolated charges e separated by 0.7 mir in water at 25 “C, and we assume that the line charge q around the disk is sufficiently low to avoid condensation.

We draw the reader’s attention to finite element solutions of the nonlinear Poisson-Boltzmann equation around a charged disk of finite thickness, obtained by Secor and Radke12 and Chang and Sposito.13

B. The perturbed charge cloud

When an electric field Em is applied, the charge cloud around the particle will’ be perturbed, both by the direct electrical forces acting on the ions, and by the motion of the fluid. Ions are convected with the fluid velocity u, and move relative to the fluid under the influence of electric fields and thermal diffusion, and hence the conservation equation for the ith ionic species, in steady state, is

V.[n’u-w’(kTVn’+ezgz’V$)]=O, (11)

where o’ denotes the mobility of the ith species. We follow Saville’ and nondimensionalize potentials by

kTle, lengths by a, velocities by e(kTle)2/pa, and mobili- ties by 0’. We assume that the potential aE” which chatac- terizes the applied field is small compared to the equilibrium zeta potential $. at the surface of the particle, so that p= aeE”lkT~ecjlolkT, where erClolkT has already been assumed small in order to derive the linearized Poisson- Boltzmann equation (3). We use the dimensionless field strength p as the basis for a perturbation expansion:

ii=pii,+--- ) (124

~=~o+p&+- ) (12b)

nLni,+pn+... , WC)

where the subscript 0 refers to the equilibrium cloud, and the caret A denotes a nondimensional quantity. The steady-state ion conservation equation, correct to O(p) becomes

P,G~.vn~=G’V.(z~n~v~~+z~n~V~o+Vn”l), (13)

where the P&let number P,= ekTfe2puw0 is a measure of the ratio of ionic convection to diffusion. Since the nondimensional equilibrium potential a0 has been ‘assumed to be small, (13) reduces’to

(14)

The (dimensional) boundary conditions at infinity are

n&O, (154

PA --Em-r. (1%)

We assume that no ions enter or leave the surface of the particle:Hence

n.(kTVnifeziniV+)=O, (16)

where n is-the normal to the particle surface.& O&I), and assuming +o&l, this zero flux boundary condition becomes

n*V(n~+zp~&>=O. (17)

Multiplying Eq. (14) by e2zi and summing over i, we obtain, at O(p)

v”,$1=0, W3)

and

(20)

is the (nondimensional) perturbation to the charge density p. The boundary conditions for (18) are obtained in similar fashion from Eqs. (15) and (17):

k-- iC.f; as r-w, (214

n.Vil=O on the particle surface, @lb)

where & is a unit vector in the direction of the applied field E”. The (dimensional) potential

X=$f’1+p+K2 (221

is thus obtained by solving Laplace’s equation for the potential around an insulating particle in a uniform field.

The solution of Laplace’s equation for the potential around a disk is straightforward if the applied field E” is parallel to the surface of the disk; since an infinitesimally thin disk does not perturb the electric field. Hence

x(P,z)= -Em-r (edgewise motion). ’ (23)

If the applied field Em is normal to the surface of the disk, the field may be obtained by means of Hankel transforms,14 and is

2E” - x(r,z)= -E”z- 7

I o Ct- 2 sin(at)-at-I-cos(at)]

Xexp( -tz)J,(rt)dt, 220. (24)

Phys. Fluids, Vol. 7, No. 4, April 1995 J. D. Shewood and H. A. Stone 699


Integrating by parts, scaling all lengths by a, making the transformations (a conversion to oblate spheroidal coordinates)

and using Sec. 6.752 of Gradshteyn and Ryzhik,r’ we obtain

2E” ~(r,z)=-E”z+ 7 (z cot-’ A-i)(broadside motion).

i26) We shall later require

1 sin t-cos t]

X exp( - tz)Jo(rt)dt

(274 2E”

i

A Em+ 7 cot-t h-m ,

i

JX 2E” 0) -=- I dr n 0

[t-l sin t-cos t]exp( - tz)J,(rt)dt

2E” c( 1 - c2) =- 7r r(h2f c2) ’ cm

C. Fluid motion and an application of the reciprocal theorem

where n is the outward normal to the surface of the particle and V is again the entire fluid volume outside the particle. The equilibrium contribution poV& is balanced by the pressure acting over the surface of the particle, and as in the Stokes equation (29), the leading-order contribution to the direct electric force may be expressed as

The Stokes equations are modified by the presence of an electric force pE acting on the fluid. Expanding in powers of /3, we obtain

Fe= I

poVx dV. (34) V

+o(P2L (28)

where we have made use of the identity po= - •24~. Hence the Stokes equations become

J&u-vp-pov,y=o, (2%

where the term V($&&-@r~&o) in (28) has been incorpo- rated into the pressure p. The effect of this additional pressure on the particle is canceled by terms in the direct electric

I stress (33) discussed below. I

If the charge cloud is large compared to the size of the particle, we may approximate Vx as -Em, and hence F,=QE”; this simplification was used by Sherwood.6 Combining the fluid ‘stress (32) and the direct electric force (34), the total force balance on the particle may be written as

I poVx.(I-G)dV-R.U=O.

V (35)

Note, that the fluid motion caused by direct electrical forces acting on the fluid, and that caused by motion of the particle, will Freate additional deformation of the charge cloud, but, as seen from Eq. (13), this deformation will be O(li,P,q$,), and may be neglected when the potential +. is small.

The force on the particle may now be evaluated by ! means of the reciprocal theorem for Stokes flows.r6 Details ! I are given by Sherwood6 and by Teubner.7 If the (uncharged) I particle translates with velocity U, it exerts a force

F Stokes=R.U on the fluid where R is the resistance tensor for the particle. The corresponding fluid velocity is

u(r)=G(r)eU, (30)

where G(r)’ is a second rank tensor, with G=I (the identity tensor) on the surface of the particle. The velocity far from the particle will be that due to a Stokeslet of strength FStokes , and hence, for Irj%a,

G(r) - J(rlR

Equation (35) gives the electrophoretic translation velocity of an arbitrarily shaped particle accounting for the influence of the charge’cloud. Knowledge of G and R for a given particle shape, along with the corresponding solutions for x and (PO, allow calculation of U. We now turn the explicit form of G for disk-shaped particles.

Several authors have studied Stokes flow around a disk: Ray17 and Gupta18 considered the case of broadside motion and Ray17 and Davis*’ considered the case of edgewise motion; results are conveniently summarized by Tanzosh.20 The components of the tensor G (for ~20) that we require, expressed in the spheroidal coordinates (&Q, are

G,,=i I

2 zX(l+?) -z sin tJl(rt)exp(-tz)dt= -

0 m r(X2+12) ’ where (36a)

J(r)= & I+ $ , i i (31)

and r is the position vector relative’ to the center of the particle.

The effect of the electric body force per unit v$ume, -poVx, in the Stokes equation (29) will create a disturbance flow in the fluid surrounding the particle, which results in a force on the particle

F cloud= - I

~0vx.G dV> (32) V

where the volume integral extends over the entire fluid volume V outside the particle.

The electric force F, acting directly on the particle may be obtained by integrating the Maxwell stress tensor m over the surface S of the particle, to obtain7

Fe= Isman dS= j”pV4 dV,

700 Phys. Fluids, Vol. 7, No. 4, April 1995 J. D. E$erwood and H. A. Stone


Jo(rt)exp( - tz)dt

exp( - tz)[( 1 + tz)Jz(rt)cos 2 0

Wb)

+ (3 - tz)Jo(rt)]dt

2

i Pu-$) u2 =-

37r (l-A”)(A%-p) cos 2e+3 cot -l +Tqz 1

SE($) ~0s 20+1’-$~) xx xx * (36~)

The well-known resistance coefficients characterizing disk translation in a Stokes flow are (e.g., Happel and Brenner16)

R,,=y pa, R,= 16pa.

Substituting G (36), p. (5), and V,y (27) into the force balance (35), performing the angular integration, and scaling lengths by a and x by aE”, the electrophoretic velocity of a disk in an electric field normal to the plane of the disk is (broadside motion)

u broad= QE;;‘2 6,6,$0( 2 (Gzz-1)

If the electric field is parallel to the face of the disk, we obtain the edgewise translational velocity

u 3&E”(aK)” m

edge= ------I I m -

16rrpa qSo( 1 - GFX))r dr dz, (394

r=O x=0

= g [ l-2(aKj~~~o~~o&oG~)r dr dz),

V W where

sin t exp( - tz)Jo(rt)dt

(40)

For arbitrary values of a~ we must evaluate these expres- sions numerically. However, for the limiting cases of thin and thick double layers, it is possible to obtain analytical estimates, as shown in the next section.

iii. AsYMPToTic RESULTS FOR aKs1 AND aKe1

A. Thin double layers-Edgewise motion with a uniform surface charge

When the disk is uniformly charged and the double layer thin, KaSl, we may approximate the equilibrium potential by (8), except over a negligibly small region at the edge of

the disk, and the integration in (39) need only be taken over the region Osrsl. Performing first the integration over z, and then the integration over r, we obtain

$oGit)r dr dz

sin t (2tf3aKj =-

t tt+aK)2 Jdrtb dr dt

2 sin t (2tf3afc)

=5&G -P- (t+aKy J1(t)dt

1 8 = 2(aK)2- 3T(aK)3+O(aK)-4,

where we have used the results21

(41)

I m sin t

o 7 JI(tjdt= ;, (42)

Hence, to leading order when a~%l,

u QswF @o em e&e= zpra$ = 7 (43)

which is Smoluchowski’s result; note that we have used the fact that a uniformly charged surface in the thin double layer limit may be characterized by a uniform zeta potential ~=cT/KE=&&?‘~~~KE.

The results of the next section (III B) suggest that edge corrections to (43) will be O(a~)-l’ smaller.

B. Thin double layers-Edgewise motion with a uniform edge charge

We next consider the limit a~%-1 for the edgewise motion of a disk with an edge charge. The dominant contribution to the integral comes from the vicinity of the edge and so we carefully consider the neighborhood of (r,z)=(l,O) where the spheroidal coordinates (&h) are small, i.e., 541, hel. Using a local system of cylindrical coordinates (R,cr) based on the edge of the disk, with the angle a=0 corresponding to the plane z=O outside the surface of the disk, we find

R cos a=r-l==$(A2-12), (444

Pb)

and hence

A”=R(l+cos n)=2R cos2(42), (454

~2--R(1-cos a)=2R sin”(42). Wb)

The dominant contribution to G$j in the neighborhood of the edge is

Phys. Fluids, Vol. 7, No. 4, April 1995 J. D. Sherwood and H. A. Stone 701


(46)

If a~%=l, we may neglect curvature of the edge charge. The potential around the edge charge of density q = QedgJ2 rra may be approximated by that around an infinite line charge:

+o=& fGda4, (47)

where K, is a modified Bessel function and R has been non- dimensionalized by the disk radius a; the corresponding charge density follows from (5).

An electric field applied parallel to the surface of the disk is not perturbed by the disk, so Vx=-E”. Inserting these asymptotes (46), (47) into the force balance (35) we obtain

3+aUedge

3 =2ra3 *da2

-7I

XKo(aKR)E” k (2R)“2

Xcos( ;)[ 3+&( $1.

Integrating first with respect to the angle cr gives

32pa lJedge 8Ov2 m =-

3 9%. a3K2qEm

I R312Ko(aKR)dR.

0

(48)

(49) I

This integral may be evaluated analytically using the result (Watson,“’ p. 388)

I =y+K,(y )dy = 0

2’LT( y)r( $Lj. (ziO)

Hence the electrophoretic velocity in the limit a KS 1 is

u edge = 3Q,,@” WZ~(~~12

32pa 9?r2(aK)1iZ’ I where I’(5/4)=0.9064. Comparing (43) and (51) we observe I that for edgewise motion with an edge charge distribution the

electrophoretic translation speed is higher than for a surface charge distribution. The electrical forces acting on the charge cloud (with total charge -Qj tend to oppose the motion of the particle. These forces are transmitted, via fluid stresses, to the surface of the particle, and this coupling is weaker at the edge of the disk than over the interior.

If the disk is uniformly charged, as in Sec. III A, the amount of charge within a distance 0 (K-~) of the edge of the disk is O(Q&aK), and this, by (51), will create an electrophoretic velocity OIQsu,&“lpa(a K)“‘“]. Hence edge corrections to (43) will be O(~K)-” smaller.

C. Thin double layers-Broadside motion with an edge charge

For the limit a~%1, the third case to consider is the broadside motion of the disk with an edge charge distribution. We proceed in a manner similar to the calculations presented above. In particular, we find that in the neighborhood of the corner

(52)

and

JX 2.E” c( 1 - 5’) 2E” sin( a/2)

dr= i-r i-(X2+&--y (2R)l” ’

The hydrodynamic coupling tensor has components

and

(53)

on= 22X(1 -572 2 m-(X2+ J2)

-; (2R)l” cos’ ; sin ; . 0 0

(55)

The broadside electrophoretic velocity Ur,r,& is given by the force balance (35)

16 TPa Ubroad

=27ra3 2 4E” (Y

da 2 Ko(aKR) ---J cos’ - 0 2

664 = z (aKj2Qedg~m~om~~(a~)R dR, (56b)

and so using (50) we arrive at

Ubroad= Qed,&” 8 r2pa ’

(57)

which is independent of a~.

Note that we do not present any asymptotic approximations for the broadside motion of a uniformly charged disk. The standard thin double layer analysis for the interior of the disk will lead to Smoluchowski’s result (43). The edge cor- rection will correspond roughly to (57), with an edge charge of order Qedge-Q,,&aK, and thus cannot be neglected. Hence we expect Ubroad-CQ,&“/2pua2tc, with some un- known numerical factor C.

702 Phys. Fluids, Vol. 7, No. 4, April 1995 J. D. Sherwood and H. A. Stone


D. Thick double layers

If the Debye length K-~ is large compared to the particle dimension a, the dominant contribution to the volume integral in (35) comes from the O(K-~) volume of the charge cloud. On this length scale the particle acts as a point charge, for which the potential of the surrounding equilibrium charge cloud is

4 =Q exp(-Kr) (+.I) 0 4 mr

and acts as a point Stokeslet, so that the tensor G is given by (31). The solution of the Laplace equation (18), (21) for the field around an insulating particle consists of the imposed applied field -Em together with dipole and higher multipole corrections, which we can neglect as we are only interested in the dominant contribution to the integral. Inserting these approximations into the force balance (35), we obtain, for broadside motion

- ._x___ Ubroad -

and for edgewise motion

u (60)

It is straightforward to apply this thick double layer analysis to the case of a disk with a uniform surface potential (rather than a uniform charge density). Far from the particle, the potential will still have the form (58); only the quadrupole and higher multipole terms in the expansion of the potential around a disk at uniform potential will differ from those around a disk with a uniform surface charge density. The total charge on a disk at potential I++~ is22*27

Q=8eatio[ ,+saK+( -$-i)(aK)2+eem], aKe1.

(61) Hence at constant potential

(62)

and

U edge 2aK

l+p,fO(a/c)2 , aKe1. (63)

The above asymptotic results for a~%1 and a~<1 provide useful checks on the numerical evaluation of the electrophoretic velocity reported in the next section.

IV. NUMERICAL RESULTS

The electrophoretic velocities for broadside and edge translation were determined by numerically evaluating Eqs. (38) and (39), respectively. A rectangular grid, centered on the disk, subdivided the TZ plane. The two-dimensional integration over each rectangular element was performed using the IMSL routine DQANDO, which utilizes an iterated application of Gaussian quadrature formulas; typically an absolute

0 G-

5. 2't LlJ -0.5

&ol 2

- -1 s 2

-2.l I -2 -1 0 1 2

lo&&K)

FIG. 2. The nondimensional electrophoretic velocity for edgewise translation, Uedge 16/*a/QE”, as a function of UK. Solid curves are the numerical results for a uniform surface charge and a uniform edge charge. Dashed curves are the asymptotic approximations (43),(51),(60) derived in Sec. III.

accuracy of 10m8 was specified for each integration. The evaluation of several of the intermediate integrals in the kernel [e.g., (27)] in terms of the spheroidal coordinates (A,& which are easily evaluated in terms of (r,z), greatly simpli- fied the numerical work. The integration in the rt plane was terminated at distances large compared to (a~)-‘; integration over a larger distances was performed and results reported here are accurate to within a few percent. The equilibrium potential 4. requires an infinite integration with an oscillatory kernel involving a product of Bessel functions and the routines described and implemented by Lucasz4 were used to calculate these integrals to an absolute accuracy of 10-8.

The computations for thin double layers (a KS 1) are dif- ficult numerically. In this limit, the gradients of the potential $. are typically large near the edge of the disk; a mesh finer than (aK)-’ was set up in the neighborhood of (~,z)=(l,O) in order to resolve these significant contributions to the integrals.

In Fig . 2 we show results for the electrophoretic velocity, scaled by QE”/l6pa, for edgewise translation. Both the case of a uniform surface charge and a uniform edge charge are shown with solid curves. The dashed curves are the asymptotic results described in Sec. III for both the thick (60) and thin (43), (51) double layer limits. For aKC0.1, a~>5 (edge charge), and a~>50 (surface charge), the numerical results are within about ten percent of the analytical predictions. For a ~<l, where the charge cloud begins to get larger than the disk, the translation speed is the same for surface and edge charges of the same magnitude. Also, for the same double layer thickness (with aK>l) and total charge on the disk, the disk with surface-distributed charge translates more slowly than the disk with edge-distributed charge.

In Fig. 3 we consider the case of broadside electrophoretic motion, with velocity again scaled by QE”/16,ua, for both a surface charge distribution and an edge charge distribution. Dashed curves indicate asymptotic approxima-



0.5 ----- T-

o -- P 3.

sfl, 1 w -OS-- eQ

sl 0 -1 -- .F

-21. : I -2 -1 0 1 2

log,,(alc)

FIG. 3. The nondimensional electrophoretic velocity for broadside translation, hoad 16@QE”, as a function of UK. Solid curves are the numerical results for a uniform surface charge and a uniform edge charge. Dashed curves are the asymptotic approximations (57),(59) derived in Sec. III, and in the case of broadside motion for a~%=1 the asymptote is given by (64) with the constant C=O.226.

tions for thick double layers (59), and for a thin double layer around a disk charged at its edge (57). These agree well with the numerical results. It was argued in Sec. III C that the broadside velocity of a uniformly charged disk would vary as

u ~roa~-CQsur~Emi2~~a2K, 64)

and we may use the numerical results shown in Fig. 3 to establish C -0.226. This is smaller than the value C = 1 pre- dicted by Smoluchowski for edgewise motion (43).

Comparing Figs. 2 and 3 we see that as the double layer gets vanishingly thin (a K-W) the disk will move broadside since any edge charge leads to a finite broadside translational speed (57), while the other cases have velocities which van- ish as a K increases.

Finally, we examine the ratio of the edge charge to surface charge Q,,./Q,, (CO) at which the electrophoretic velocity is zero. Results for both the transverse and edgewise

0-l I -1 0 1 2

log,,(aK)

FIG. 4. The charge ratio IQeds$Qsurfl at which the electrophoretic velocity is zero. The solid curves denote numerical results for edgewise and broadside motion and the dashed curves are based upon the asymptotic approximations developed in Sec. III.

motions of the disk are shown in Fig. 4. The principal con- clusion to be drawn from this figure is that larger ratios of edge to surface charge are necessary to effect the same change in electrophoretic velocity for the case of edgewise motion as compared to broadside motion.

V. DISCUSSION

The standard representation of a Na-montmorillonite particle is a flat plate with a uniform negative surface charge rr, typically”5,26 of order 0.1 C rr-‘. In the absence of a Stern layer of adsorbed ions, linear Poisson-Boltzmann theory predicts a surface potential $bo=(TIc~; the permittivity E= eOer, where E,, is the perrnittivity of free space and er is the relative permittivity of the fluid. Taking e,=gO (water) and ~~~~10 nrn (l-l electrolyte at 1O-3 moljl) we obtain $a= -1.4 V, or e$,JkT= -55. Nonlinear theory” predicts a nondimensional potential e JrolkT =2 sinh-‘( mei2 EKkT) =-8, or $a=-200 rnV. Thus the linear theory of Sec. II would at first sight appear not to be appropriate for montmorillonite.

In practice, however, the surface charge c~ is shielded by counterions which are to some extent bound to the surface. The degree of binding is ion-specific, as is observed in measurements both of electrophoretic velocities28 and of the forces between particles.‘6.29 Such ion binding is also pre- dicted by Monte Car10~~ and molecular dynamics31 simula- tions of the clay-water interface. Measured zeta potentials usually have magnitudes in the range O-100 mV. Full nonlinear computations of the electrophoretic velocity of spherical particles show32 that linear theory is a good approximation up to nondimensional surface potentials of order 4. The linear analysis of Sec. II should therefore provide a reason- able approximation. When the Debye length K-~ is small compared to the size of the particles (i.e., in the limit of a point particle), nonlinear computations indicate that the linear analysis holds to much higher potentials. This gives some reassurance that the high potentials associated with the edge charge q will not invalidate the linear analysis presented here.

The edge charge of a clay particle depends upon the pH, and electrophoretic velocities can increase by 50% or morerp2 as the pH increases to values above 10. However, electrophoretic velocities which are independent of pH have also been reported,33Z34 and so have apparent zeta potentials which vary little with electrolyte concentration.33 It has therefore been suggested2895 that a constant potential boundary condition might be more appropriate. These differences between the various experimental observations make it hard for us to compare theory against experiment. If the montrno- rillonite particles are large, the ratio of the edge charge Q edge=2maq to the surface charge QsUrf=2ra20 will decrease, thereby reducing the effect of the edge. Particle size therefore plays an important role.

The results presented in Sets. III and IV indicate that as the double layer becomes thinner, the edge charge gives a greater contribution to the electrophoretic velocity than the surface charge; the edge charge electrophoretic velocities are O(4 1’2 larger for edgewise motion, and O(a K) larger for broadside motion. We may thus conclude that in the limit

704 Phys. Fluids, Vol. 7, No. 4, April 1995 J. D. Sherwood and H. A. Stone


atc--+w, the disk will move broadside if the edge charge is nonzero. The electric field is singular at the edge of the particle, and interacts strongly with the edge charge. However, in practice, our analysis is no longer appropriate when the Debye length K-~ becomes so small that the thickness of the clay platelet may no longer be neglected; for sufficiently thin double layers the analysis of Fair and Anderson4 is more appropriate. They specified the zeta potential as a function of position over the surface of a spheroid, and assumed that the double layer was everywhere thin compared to the particle dimension.

Yoon and Kim3 studied spheroids with a uniform zeta potential @a, but with arbitrary double layer thickness. Our analysis is therefore similar to theirs in the limit in which an oblate spheroid becomes a disk. When the double layer is thin, there will be no difference between the condition of constant surface charge density (+ adopted here, and that of constant zeta potential, except in a region O(K-I) around the edge of the particle. If Yoon and Kim had studied sufficiently thin double layers, their results, like those of Fair and Ander- son, would ultimately agree with Smoluchowski (43). This limit is not reached in our own analysis of broadside motion of uniformly charged particles, as the double layer is never thin compared to the plate thickness, and hence C # 1 in (64). In this case, the charge at the center of the disk is shielded from the broadside electric field. It contributes little to the electrophoretic velocity, which depends upon the detailed distribution of ions and charge around the edgy of the disk, where the field is high. Two limiting processes may be considered: that in which the double layei’thickness around an ablate spheroid decreases to zero and that in which the spheroid approaches a flat disk. Clearly the two processes do not commute.

In the limit of a thick double layer (a ~41), the charge distribution at constant potential is (highly) nonuniform, and the results of Yoon and Kim differ from ours. In particular, at constant surface charge, we predict that the electrophoretic velocity’ decreases as a K increases, for both edgewise (60) and broadside motion (59). At constant potential, broadside velocities decrease (62), but edgewise velocities increase (63), as found numerically by Yoon and Kim.

ACKNOWLEDGMENTS

We thank Dr. S. K. Lucas for making available programs for integrating the Bessel function integrals and Dr. J. P. Etnzosh for helpful discussions regarding solutions to Stokes equations for the disk geometry. HAS thanks the National Science Foundation for support via a PYI Award (CTS- 8957043).

‘S. L. Swartzen-Allen and E. Matijevif, “Colloid and surface properties of clay suspensions II. Electrophoresis and cation adsorption of montmorillo&e,” J. Colloid Interface Sci. 50, 143 (1975).

‘B. Rand, E. PekenC, J. W. Goodwin, and R. W Smith, “Investigation into the existence of edge-face coagulated structures in Na-Montmorillonite suspensions,” J. Chem. Sot. Faraday Trans. I 76, 225 (1980).

sB. J. Yoon and S. Kim, “Electrophoresis of spheroidal particles,” J. Col- loid Interface Sci. 128, 275 (1989).

‘M. C Fair and J. L. Anderson, “Electrophoresis of nonuniformly charged ellipsoidal particles,” J. Colloid Interface Sci. 127, 388 (1989).

‘D. C. Henry, “The cataphoresis of suspended particles,” Proc. R. Sot. London Ser. A 133, 106 (1931).

6J. D. Sherwood, “Electrophoresis of rods,” J. Chem. Sot. Faraday Trans. II 78, 1091 (1982).

7M. Teubner, “The motion of charged colloidal particles in electric fields,” J. Chem. Phys. 76,5564 (1982).

t

sR. W. O’Brien and D. N. Ward, “The electrophoresis of a spheroid with a thin double layer,” J. Colloid Interface Sci. 121, 402 (1988).

‘D. A. Saville, “Electrokinetic effects with small particles,” Annu. Rev. Fluid Mech. 9, 321 (1977).

“G. S. Manning, “Limiting laws and counterion condensation in polyelec- trolyte solutions. I. Colligative properties,” J. Chem. Phys. 51, 924 (1969).

‘rM. Fixman, “The Poisson-Bbltzmann equation and-its application to uolvelectrolvtes,” J. Chem. Phvs. 70, 4995 (1979).

t’k - ._,.

B Secor and C. J. Radkd,.“Spillover of the diffuse double layer on . . Montmorillonite particles,” J. Colloid Interface Sci. 103, 237 (1985).

13F.-R. C. Chang and G. Sposito, “The electrical double layer of a disk- shaped clay mineral particle: effect of particle size,” J. Colloid Interface Sci. 163, 19 (1994).

141. N. Sneddon, The Use of IntegVal Transforms (McGraw-Hill, New York, 1972), p. 350.

‘s1 S Gradshteyn and I. M. Ryzhik, Tables of Integrals, Series and Products . . (Academic, New York, 1965).

16J. Happel and H. Brenner, Low Reynolds Number Hydrodynamics (Marti- nus Niihoff, Amsterdam, The Netherlands, 1983). --I-

r7M. Ray, “Application of Bessel functions to the solution of problems of motion of a circular disk in viscous liquid,” Philos. Mag. 21, 546 (1936).

r8S. C. Gupta, “Slow broad side motion of a flat plate in a viscous fluid,” Z. Angew. Math. Phys. 8, 257 (1957).

19A. M. J. Davis, “Some asymmetric Stokes flows that are structurally similar,” Phys. Fluids A 5, 2086 (1993).

2oJ. P. Tanzosh, “Integral equation formulations of the linearized Navier- Stokes equation: applications to particle motions in rotating viscous flows,” Ph.D. dissertation, Harvard University, 1994.

‘lG. N. Watson, Theory of Bessel Functions (Cambridge University Press, Cambridge, 1945), p:403.

‘*C. G. Phillips, “The steady, diffusion-limited current at a disk microelec- trode with a first-order EC! reaction,” J. Electroanal. Chem. 296, 255 (1990).

“M. A. Bender and H. A. Stone, “An integral equation approach to the study of the steady state current at surface microelectrodes,” J. Electroa- nal. Chem. 351, 29 (1993).

?S. K. Lucas, “Evaluating infinite integrals involving products of Bessel functions of arbitrary order,” J. Comput. Appl. Math. (in press, 1995).

=H. Van Olphen, An Introduction to Clay Colloid Chemistry, 2nd ed. (Wiley, New York, 1977).

26S. D. Lubetkin, S. R. Middleton, and R. H. Ottewill, “Some properties of clay-water dispersions,” Philos. Trans. R. Sot. London Ser. A 311, 353 (1984).

s7W B Russel, D. A. Saville, and W. R. Schowalter, Colloidal Dispersions (Cambridge University Press, Cambridge, 1989).

?S. E. Miller and P. I? Low, “Characterization of the electrical double layer of montmorillonite,” Langmuir 6, 572 (1990).

2VR. M. Pashley, “DLVO and hydration forces between Mica surfaces in Li+, Na+, K+ and Cs+ electrolyte solutions: a correlation of double-layer and hydration forces with surface cation exchange properties,” J. Colloid Interface Sci. 83, 531 (1981).

‘ON. T. Skipper, K. Refson, and J. D. C McConnell, “Computer simulation of water in 2:l clays,” J. Cliem. Phys. 94, 7434 (1991).

“K. Refson N. T. Skipper, and J. D. C. McConnell, “Molecular dynamics simulation’of water mobility in smectites,” in Geochemistry of Clay-Pore Fluid Interactions, edited by D. A. C. Manning, P. L. Hall, and C. R. Hughes (Chapman and Hall, London, 1993), pp. 62-77.

32R. W. O’Brien and L. R. White, “Electrophoretic mobility of a spherical colloidal particle,” J. Chem. Sot. Faraday Trans. II 74, 1607 (1978).

m1. C. Callaghan and R. H: Ottewill, “Interparticle forces in montmorillonite gels,” Faraday Disc. Chem. Sot. 57, 110 (1974).

34D. Heath and Th. F. Tadros, “Influence of pH, electrolyte and Poly(Vinyl Alcohol) addition on the rheological’characteristics of aqueous dispersions of sodium montmorillonite,” J. Colloid Interface Sci. 93, 307 (1983).

35D. Y. C. Chan, R. M. Pashley, and J. P. Quirk, “Surface potentials derived from co-ion exclusion measurements on homoionic montmorillonite and illite,” Clays Clay Miner. 32, 131 (1984).

i



electrophoresis of a thin charged disk - princeton universitystonelab/publications... ·...

Documents